2 Answers
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The Casimir effect is analogous to gravity in only one way--- it has a negative energy which varies as a power of the distance. In all other ways, it is different. The power-law is different, the cause is different, it is a quantum effect, not a classical effect, and the mediator is the electromagnetic field, not the gravitational field.

Negative energy is not surprising--- it just means that there is an attractive force (and zero potential at infinity), so that the potential energy is less than when the objects are far apart. In this respect, both Casimir and gravity have negative energies. This energy is an energy difference between two configurations.

The total energy in the Casimir system is still positive, it is just less than if you take the two plates to infinity. Similarly, the total energy in a gravitating system is also positive, but when objects are close, it is less than the energy than when they are far. This is the negative energy in gravity.

For the case of Casimir forces, there are two ways to view it. The more primitive way is to consider the fluctuating polarizations in a quantum system that lead to attractive forces. These attractions come in two types, depending on the ratio of the distance to (c times) the orbital period of the electrons (the difference in energy levels converted into a time):

London forces: when the atoms are significantly closer together than the wavelength of light that they would emit between the first excited state and the ground state, the electromagnetic interaction is instantaneous Coulomb interaction. The Coulomb interaction correlates the charge fluctuations on the two atoms so that they tend to be polarized in the same direction, and this correlated polarization leads to a $1/r^6$ attractive force between points separated by a distance r, or a $1/r^4$ force between two sheets separated by distance r.

Van-Der-Waals forces: When the atoms are further than the typical wavelength of light, the electrostatic force is no longer instantaneous, and the correlated force is weaker than you get from the statically correlated system. The force falls off as $1/r^7$ for points, and $1/r^5$ for sheets. This calculation is harder than the previous one, but when Casimir calculated the effect, it revealed itself to be more universal than the London-forces, it looked like it doesn't care as much about the energy levels, that it only depends on polarizability of each object separately.

Bohr suggested to Casimir that the universality could be understood because the attraction was due to the correlations in the quantum field surrounding the materials, and that he should calculate the force only by the vacuum energy in the surrounding field modes. This calculation works, and produces the correct form of the Van-Der-Waals force, and explains the universality.

But the effect is not hard to understand--- the Casimir effect is the Van-der-Waals attraction between two plates separated by a distance large enough so that retardation effects are important. In Dirac gauge, you have electrostatic forces and photons, and the electrostatic force is not important, and the photon vacuum energy reduction is just due to the different photon modes that are allowed at different distances.

This is not mysterious, and it shouldn't be presented as mysterious. But it is a source of endless pseudoscience about machines that extract vacuum energy. The Casimir force doesn't allow you to extract any more energy than any other attractive effective interaction.

Not trying to further the crazy wordage in my first link just used as referance to the attraction existing. Slowly building mathematical proof for my very original TOE without trying to ask stupid questions pure happenstance that perticular site came up instead of arxiv.org/abs/1105.4714
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ArgusMay 26 '12 at 4:47

To simplify your answer for my personal understanding this force is opposite gravity in the sense of magnitude as in as the plates move apart there is less "pressure" outside to the point of infinity(so one plate at one " edge" of "space" and other plate at other edge as there is "nothing" beyond the plate there is no outside pressure and only pressure between the forces.
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ArgusMay 26 '12 at 5:19

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Uhh, just to remark on your last paragraph, there are two valid views to the Casimir force: 1) van der Waals interaction (fluctuations in the material), and 2) zero-point energy (fluctuations in the vacuum) ... (2) is not pseudoscience, nor is (1) any more valid.
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Chris GerigMay 26 '12 at 9:19

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@ChrisGerig: Right--- they are equivalent points of view, but for some reason pseudoscientists seize on the vacuum energy calculuation as a demonstration that we can "harnass the infinite vacuum energy to do useful work" and nonsense like this. The alternate picture makes it clear that infinite energy cannot be extracted, since it is the material fluctuations which are responsible for lowering the vacuum energy, and then it is not magical seeming anymore.
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Ron MaimonMay 26 '12 at 19:55

@RonMaimon: From my point of view the infinite energy is already put two work makeing the fluctuations occur therefore observable results since this is being used to maintain the fluctuations in material they can not be used effectively for anyything else. Common sense infinite energy put to infinite work is a little busy already..
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ArgusMay 27 '12 at 18:51

Since relativity explains more what gravity does, but not necessarily what gravity is, your question is a valid one. Explaining gravity as some kind of residual EM force (Van der Waals, Casimir, and many other variations) actually has a long history, but they have all failed.

They essentially give solutions with too large of a negative power term "n" and (something often not stressed) too large of the constant term "A" in the force law F=A/r^n. Given the very tiny A term one would expect for gravity if such could be calculated (gravity is 10^35 times weaker than the EM force) suggests possibly our math is just not powerful or delicate enough to really tease out such a term if it were to exist. Even calculations for Van der Waals, Casimir, etc are quite difficult and involve questionable mathematics as well (summing infinities, perturbative expansions, etc).

The jury is probably out on a question such as yours. Recent results that "tunneling" of Casimir forces in 1D can result in smaller n (1/r^3) are possibly of interest to you, as well as papers prints in arXiv discussing space as a type of entanglement of the measurement process and gravity as an entanglement entropy. In addition, reading the introductory chapters on Feynman's gravitational lecture series also discusses some of these topics as well.